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Keywords:

  • millimeter wave radio propagation;
  • rain attenuation prediction;
  • differential evolution algorithm

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Work Related With the Experimental Setup
  5. 3. Formulation of the Proposed Model
  6. 4. Differential Evolution Algorithm
  7. 5. Numerical Simulations and Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[1] The principal objective of a rain attenuation prediction method is to achieve acceptable estimates of the attenuation incurred on the signal due to rain. In this paper, a differential evolution (DE) based model for predicting rain attenuation in a terrestrial point-to-point line of sight (LOS) link at 97 GHz is proposed using previously available experimental data obtained in the southern United Kingdom. Rainfall rate and percentage of time are used as input data in the proposed prediction model. Excellent agreement between the experimental data and the model output indicates that the presented DE based method may efficiently be used for accurate prediction of the rain attenuation levels.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Work Related With the Experimental Setup
  5. 3. Formulation of the Proposed Model
  6. 4. Differential Evolution Algorithm
  7. 5. Numerical Simulations and Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[2] It is well known that the rain attenuation caused by electromagnetic scattering and absorption is the most pronounced effect on terrestrial communication systems operating at frequencies above 10 GHz. This propagation effect limits the path length of radio communication systems and restricts the use of higher frequencies, both for line-of-sight (LOS) microwave links and for satellite communications. The knowledge of the rain attenuation from rainfall rate is extremely required for the design of a reliable communication system at a particular location [Crane, 1980; Green, 2004]. Because the utilization of upper frequency bands increases rapidly due to the application in radar, wireless and satellite communications, the importance of rain attenuation is a common subject shared by telecommunication service providers in the world.

[3] Recently, Khan et al. [1999] established a millimeter wave experimental equipment between the University of Portsmounth and Fort Widley for a 97 GHz point-to-point LOS terrestrial link. This link was realized in an urban environment on a path length of 6.526 km and operated over a period of 12 months from February 1999 to January 2000. In Khan et al. [2003], experimental rain attenuation results recorded from this link have been presented by the same authors, and the results have been also compared with both the ITU-R model [International Telecommunication Union (ITU), 1992] which is the leading rain attenuation prediction model and the Goddard et al. model introduced by Goddard and Thurari [1997]. As reported by Khan et al. [2003], the agreement between the measured rain attenuation values and those computed by the two above mentioned prediction models for this link was not entirely satisfactory.

[4] In a more recent work [Develi, 2007], an artificial neural network (ANN) based method for predicting rain attenuation in terrestrial point-to-point LOS links at 97 GHz has been proposed. In that work, very good predictions were obtained, especially for the input values that were used to train the network. However, from a modeling point of view, the major disadvantage of an ANN that of its behavior as a “black box” model, which gives the user little insight into how the network operates. The knowledge of the ANN is hidden in the interconnections between the neurons, and the inner mechanisms of the process are not easily understood.

[5] Differential evolution (DE), developed by Storn and Price [1997], is one of the excellent evolution algorithms that presents great convergence characteristics. Recently, this method has been verified as a promising candidate for solving global optimization problems. In this paper, rain attenuations in terrestrial point-to-point LOS links at 97 GHz are modeled and the parameters of the model are optimized by utilizing the DE algorithm. It is found that the prediction results of the proposed model are in excellent agreement with the measured rain attenuations.

[6] The organization of this paper is as follows: Section 2 summarizes the work related with the experimental setup reported by Khan et al. [2003]. The proposed numerical model which illustrates the nonlinear relationship between the inputs and output is presented in section 3. Section 4 describes the DE algorithm used to determine the optimal values of the parameters belonging to the proposed model. Numerical simulations and results are given in section 5. Finally, section 6 draws conclusions on the accuracy of the proposed model.

2. Work Related With the Experimental Setup

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Work Related With the Experimental Setup
  5. 3. Formulation of the Proposed Model
  6. 4. Differential Evolution Algorithm
  7. 5. Numerical Simulations and Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[7] The 97-GHz transmitter in Khan et al.'s [1999] established experimental equipment was composed of a Gunn diode oscillator. This oscillator was capable of transmitting a CW signal at 97-GHz frequency, and radiated a power of approximately 17 dBm (51 mW). The output of the Gunn diode oscillator was fed to a prime focus parabolic antenna via an isolator. The antenna was vertically polarized with 42 dB gain and a beam width of 1.5 degree. The reflector's diameter was 0.15 m. A similar antenna was used for the receiver. The receiver was a single stage superheterodyne system with a dynamic range of approximately 60 dB. A single ended mixer was used to produce an IF of 300 MHz. The mixer output was fed through an IF amplifier and a band pass filter of 30 MHz bandwidth. The transmitter and receiver systems were mounted on benches inside the cabins and with the antennas mounted indoors, the signals passed through polystyrene windows. The detailed descriptions of the experimental equipment, topographical parameters of the link, experimental geometry and data acquisition technique have been reported by Khan et al. [1999]. Also, link power budget, parameters of the link and the topographical parameters belonging to the transmitter and the receiver sites can be found in Khan et al. [2003].

[8] Figure 1 shows the rain attenuation prediction results obtained by the ITU-R model [International Telecommunication Union (ITU), 1992], the Goddard et al. model [Goddard and Thurari, 1997] and the ANN based model [Develi, 2007]. The measured values given in Khan et al. [2003] are also included in this figure. It can be noticed from the results presented in Figure 1 that the ITU-R model underestimates the rain attenuation whereas the model by Goddard et al. gives a comparatively better fit to the experimental data even though it slightly overestimates it [Khan et al., 2003]. Furthermore, as it can be seen from Figure 1, ANN based model is the most accurate model among these models [Develi, 2007]. It is well known that the prediction of rain attenuation is a very complex and difficult process. As reported in the literature, the rain attenuation poses a greater problem to microwave communication systems as the frequency of occurrence of heavy rain increases. Therefore, it is not feasible to theoretically predict the attenuation of radio waves above 10 GHz, propagating through the atmosphere, because of the complex variability of meteorological factors affecting the propagation.

image

Figure 1. Measured and predicted rain attenuation values obtained by different prediction models.

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[9] In general, the prediction models in the literature can be either empirical (also known as statistic) or theoretical (also known as deterministic), or a combination of these two [Crane, 1971, 1975; Laster and Stutzman, 1995; Bosisio and Riva, 1998; Ramachandran and Kumar, 2006]. Unfortunately, the main problem of the classical empirical models is the unsatisfactory accuracy, while the condition to implement the theoretical models is hard to meet [Yang et al., 2001].

3. Formulation of the Proposed Model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Work Related With the Experimental Setup
  5. 3. Formulation of the Proposed Model
  6. 4. Differential Evolution Algorithm
  7. 5. Numerical Simulations and Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[10] Rainfall rate and percentage of time are used as input data in the proposed prediction model. The following model is used to illustrate the nonlinear relationship between the inputs (rainfall rate and percentage of time) and output (rain attenuation):

  • equation image

where x(t) denotes the percentage of time, y(t) denotes the rainfall rate, z(t) denotes the rain attenuation, the parameters {a0, a1,…, aK} and {b1, b2,…, bN} denote the model parameters. The sum of the parameters K and N, determines the number of the input terms in the model. The model in equation (1) may alternatively be expressed in closed form as follows:

  • equation image

where the function f(·) represents the nonlinear relationship between x(t), y(t) and z(t). Given a set, which includes measurement data information, {xk(t), yk(t), mk(t)}, with k = 1, 2,…, M, where M is the number of information in the measurement set, the mean absolute model error may be defined as follows:

  • equation image

By substituting equation (2) in the above equation, we obtain

  • equation image

Since xk(t), yk(t) and mk(t) are known measurement data, the only unknowns in the above equation are the model parameters {a0, a1,…, aK} and {b1, b2,…, bN}. In this paper, the mean absolute model error given in equation (4) is used as the cost function, and values of the optimal model parameters for minimizing this cost function are determined by using the DE algorithm, which is briefly described in the next section.

4. Differential Evolution Algorithm

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Work Related With the Experimental Setup
  5. 3. Formulation of the Proposed Model
  6. 4. Differential Evolution Algorithm
  7. 5. Numerical Simulations and Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[11] Evolutionary algorithms (EAs) are heuristic methods that are based on the principles of natural evolution for solving nonlinear and multimodal optimization problems. Genetic algorithm (GA) [Holland, 1975; Goldberg, 1989] and differential evolution (DE) [Storn and Price, 1997; Price et al., 2005] are the most commonly encountered and well-established EAs used in the literature. Each of these algorithms has its own characteristics, but the GA has the disadvantage of requiring a much longer time to reach to the optimal solution of tough optimization problems, especially when the solution space is hard to explore.

[12] The DE has proven to be a highly efficient technique for finding optimum effectively with a smaller probability of falling in local optima than other EAs. The major benefits of DE are its simplicity, ease of use, and robustness. Moreover, DE uses only a few control parameters and these remain fixed throughout the entire optimization procedure. Unlike the GA that uses binary coding for representing the problem parameters, the DE algorithm uses real-code representation for the parameters to be optimized, and it operates on a population with Npop individuals. Each individual (or chromosome) is a symbolic representation of the Npar optimization parameters. Compared to GA, DE has the following advantages and can be used to overcome the drawbacks of GA. First, DE gives all parent individuals equal chance to generate the next generation, and there is no discrimination on the less-fit individuals. Secondly, the mutation is conducted by mutating the parents with population-derived difference vectors, at the beginning of each evolution loop. Thus, the destructive mutation in GA can be avoided. Finally, the competition between the individual parent and individual child takes place after crossover [Yang et al., 2003]. The flow chart of the optimization process using DE is shown in Figure 2.

image

Figure 2. Flowchart of a standard DE algorithm.

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[13] The initial population is chosen randomly within a set of given bounds if no assumptions about the solution are made. After the initialization, the algorithm goes into genetic evolution and three genetic operations, namely, mutation, crossover, and selection, are executed in sequence. Each operator prepares intermediate population for the next operation.

[14] The mutation operation is the key procedure in DE algorithm. In this work, the mutation operation, which generates a mating partner for each individual by producing a difference vector called the mutant vector, is used. The mutant vector can be written as:

  • equation image

where i, p1, and p2 are three randomly selected distinct individual indexes, n is the generation index, Pmut is the mutation factor, and the superscripts M and opt refer to the mating pool and the optimal individual in the population, respectively. Following the mutation operation, a trial vector is determined by performing the crossover operation as follows:

  • equation image

where C denotes the population of children, γ ∈ [0, 1] is a uniformly distributed random number, and Pcross is the crossover constant. The trial is then checked, and if it falls outside the prescribed bounds, it is replaced by a feasible value.

[15] As a final operation, the selection is used to create better offspring. The fitness values of the children are calculated by using the cost function given by equation (4). Each child competes with its parent, and survives only if its fitness value is better. Following this, the next round of genetic evolution begins. These processes are repeated until a termination criterion is attained or a predetermined generation number is reached.

5. Numerical Simulations and Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Work Related With the Experimental Setup
  5. 3. Formulation of the Proposed Model
  6. 4. Differential Evolution Algorithm
  7. 5. Numerical Simulations and Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[16] In this paper, we used two-level DE that finds both the function model and the appropriate coefficients belonging to the selected model. The first step is to determine the number of terms belonging to the selected model, z(t), given by equation (1). After a few runs of the DE algorithm, the values of K and N in equation (1) are found as 5 and 4, respectively. Thus, the following model that produces quite satisfactory results is obtained:

  • equation image

The key simulation parameters that must be set by the user in the DE are Npar and Npop. The search parameters of the DE algorithm are Pcross and Pmut. Proper tuning of these parameters allows good tradeoff between the global exploration and the local exploitation so as to increase the convergence speed and efficiency of the search process. In this work, the simulation parameters Npop, Pmut and Pcross are taken as 50, 0.7 and 0.9, respectively. Some basic principles and suggestions from the literature helped guide our choice of parameter values for the algorithm [Storn and Price, 1997; Price, 1999; Lampinen and Storn, 2004; Akdagli and Yuksel, 2006].

[17] Since the number of unknown parameters of the model in equation (7) is ten, Npar = 10. Thus, as the second step, the parameters (coefficients) of the model given above are then optimally determined for predicting the rain attenuations so that the output values carried out by the model converge to the measured data. This is achieved by tuning the parameters of the proposed model by the DE algorithm so as to minimize the cost function given by equation (4). Finally, the algorithm is defined to terminate when the value of objective function is less than 10−3. The time of each calculation took almost less than 35 s on a P4-2.4 GHz PC with 512 MB RAM. The parameter values found by DE algorithm are tabulated in Table 1.

Table 1. Parameter Values Determined by the DE Algorithm for the Proposed Model
Parameters of the Model (Coefficients)Values, ×10
a0−7.532
a1−19.237
a2−3.032
a346.176
a4−38.429
a59.152
b118.093
b2−7.198
b31.134
b4−0.063

[18] In order to evaluate the proposed model, we compare the rain attenuation values obtained from the proposed model with three different models: the ITU-R model, the model by Goddard et al. and the ANN based model. The prediction results are given in Table 2. The measured values are also included in the table. It can be seen that the prediction results of the proposed model are in excellent agreement with the measured values. It is also observed that the proposed prediction method reveals much better agreement with experimental data than the ITU-R model, the model by Goddard et al. and the ANN based model. Therefore, the good agreement between the presented results and the experimental data shows that the proposed model is a powerful tool for predicting the specific attenuation due to rain.

Table 2. Comparison of the Rain Attenuation Prediction Results of the Proposed Model With the Results Obtained by Different Prediction Models
Percentage of Time, %Rainfall Rate, mm/hAttenuation Due to Rain, dB
MeasuredITU-R ModelGoddard et al.'s ModelANN Based ModelProposed DE Based Model
0.16.8624.7920.0928.4724.7924.79
0.24.6518.7214.4821.6219.3818.71
0.33.5815.9011.8517.7915.9015.90
0.43.0013.6910.2515.6013.6913.69
0.52.5112.489.1313.6612.1512.47
0.62.2011.078.3012.3811.0711.07
0.71.889.867.6511.0210.119.89
0.81.659.267.1210.009.269.24
0.91.468.656.689.138.428.66
1.01.307.656.318.377.657.65

6. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Work Related With the Experimental Setup
  5. 3. Formulation of the Proposed Model
  6. 4. Differential Evolution Algorithm
  7. 5. Numerical Simulations and Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[19] Radio communication systems operating at frequencies above 10 GHz are adversely affected by rain. In this paper, a numerical method is introduced for modeling the attenuations due to rain at 97 GHz over a LOS terrestrial link situated in the southern United Kingdom. A very efficient optimization algorithm, differential evolution strategy, is employed to determine the optimal values of the parameters belonging to the proposed model. The applicability of the model is shown by comparing the calculated results with those obtained by the ITU-R model, the model by Goddard et al. and the ANN based model. It is found that the prediction results carried out by the proposed method are in excellent agreement with the measured values of rain attenuation.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Work Related With the Experimental Setup
  5. 3. Formulation of the Proposed Model
  6. 4. Differential Evolution Algorithm
  7. 5. Numerical Simulations and Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information

[20] The author thanks the anonymous reviewer and the associate editor for their thoughtful comments and helpful feedback on an early version of this paper.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Work Related With the Experimental Setup
  5. 3. Formulation of the Proposed Model
  6. 4. Differential Evolution Algorithm
  7. 5. Numerical Simulations and Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information
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  • Crane, R. K. (1971), Propagation phenomenon affecting satellite communication systems operating in the centimeter and millimeter wavelength bands, Proc. IEEE, 59(2), 173188.
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  • Develi, I. (2007), A method for predicting rain attenuation in terrestrial point-to-point line of sight links at 97 GHz, Ann. Telecommun., in press.
  • Goddard, J. W. F., and M. Thurari (1997), Radar-derived path reduction factor for terrestrial systems, paper presented at 10th International Conference on Antennas and Propagation, Inst. of Electr. Eng., Edinburgh, UK, April.
  • Goldberg, D. E. (1989), Genetic Algorithm in Search, Optimization and Machine Learning, Addison-Wesley, Boston, Mass.
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  • Holland, J. H. (1975), Adaptation in Natural and Artificial Systems, Univ. of Mich. Press, Ann Arbor.
  • International Telecommunication Union (ITU) (1992), Specific attenuation model for rain for use in prediction methods, ITU-R Recomm. P.838, Geneva.
  • Khan, S. A., A. N. Tawfik, E. Vilar, and C. J. Gibbins (1999), 97 GHz point to point line of sight radio link operating in an urban environment, paper presented at National Conference on Antennas and Propagation, Inst. of Electr. Eng., York, UK, April.
  • Khan, S. A., A. N. Tawfik, C. J. Gibbins, and B. C. Gremont (2003), Extra-high frequency line-of-sight propagation for future urban communications, IEEE Trans. Antennas Propag., 51(11), 31093121.
  • Lampinen, J., and R. Storn (2004), Differential evolution, in New Optimization Techniques in Engineering, edited by G. C. Onwubolu, and B. Babu, pp. 123166, Springer-Verlag, Berlin.
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  • Price, K. V. (1999), An introduction to differential evolution, in New Ideas in Optimization, edited by D. Corne, M. Dorigo, and F. Glover, pp. 79108, McGraw-Hill, London.
  • Price, K. V., R. Storn, and J. Lampinen (2005), Differential Evolution: A Practical Approach to Global Optimization, Springer-Verlag, Berlin.
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Work Related With the Experimental Setup
  5. 3. Formulation of the Proposed Model
  6. 4. Differential Evolution Algorithm
  7. 5. Numerical Simulations and Results
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. Supporting Information
FilenameFormatSizeDescription
rds5426-sup-0001-t01.txtplain text document0KTab-delimited Table 1.
rds5426-sup-0002-t02.txtplain text document1KTab-delimited Table 2.

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